Factorization of polynomials in one variable over the tropical semiring

TL;DR

This paper investigates the computational complexity of polynomial factorization over the tropical semiring, establishing NP-completeness results, proposing algorithms for fixed degree cases, and analyzing properties like irreducibility and tropical rank.

Contribution

It provides the first complexity classification for tropical polynomial factorization, introduces algorithms for fixed degree polynomials, and characterizes tropical divisibility and rank.

Findings

Factorization is NP-complete in general.

Algorithms exist for fixed degree polynomials.

Unique least common multiples, but not greatest common divisors.

Abstract

We show factorization of polynomials in one variable over the tropical semiring is in general NP-complete, either if all coefficients are finite, or if all are either 0 or infinity (Boolean case). We give algorithms for the factorization problem which are not polynomial time in the degree, but are polynomial time for polynomials of fixed degree. For two-variable polynomials we derive an irreducibility criterion which is almost always satisfied, even for fixed degree, and is polynomial time in the degree. We prove there are unique least common multiples of tropical polynomials, but not unique greatest common divisors. We show that if two polynomials in one variable have a common tropical factor, then their eliminant matrix is singular in the tropical sense. We prove the problem of determining tropical rank is NP-hard.